Download Chapter 5 – Simplifying Formulas and Solving Equations Section 5A

Chapter 5 – Simplifying Formulas and Solving Equations
Look at the geometry formula for Perimeter of a rectangle P = L + W + L + W. Can this formula
be written in a simpler way? If it is true, that we can simplify formulas, it can save us a lot of
work and make problems easier. How do you simplify a formula?
A famous formula in statistics is the z-score formula z 
x
. But what if we need to find the

x-value for a z-score of -2.4? Can we back solve the formula and figure out what x needs to be?
These are questions we will attempt to answer in chapter 5. We will focus on simplifying
expressions and solving equations.
Section 5A – Simplifying Formulas and Like Terms
The key to simplifying formulas, is to understand “Terms”. A “term” is a product of numbers
and or letters. A term can be a number by itself, a letter by itself, or a product of letters and
numbers. Here are some examples of terms:
12b
-11xy
W
-4
7x 2
As you can see, a term has two parts: a numerical coefficient (number part) and most of the
time a variable part (letter). Let’s see if we can separate these terms into their numerical
coefficients and variable part.
12b : We see that 12 is the numerical coefficient and b is the variable (letter) part
-11xy: We see that -11 is the numerical coefficient and xy is the variable (letter) part
(This chapter is from Preparing for Algebra and Statistics , Second Edition
by M. Teachout, College of the Canyons, Santa Clarita, CA, USA)
This content is licensed under a Creative Commons
Attribution 4.0 International license
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W: This is an interesting term as we don’t see a number part. There is a number part though.
Since W = 1W, We see that 1 is the numerical coefficient and of course W is the variable (letter)
part.
-4: This is also an interesting case as there is no variable part. This is a special term called a
constant or constant term. Constants have a number part (-7) but no variable part.
7x 2 : We see that 7 is the numerical coefficient and x 2 is the variable (letter) part.
The degree of a term is the exponent on the variable part. So since W = W 1 , W is a first degree
term. Since 7x 2 has a square on the variable, this is a second degree term. Notice the number
does not influence the degree of a term. A constant like -4 has no variable so it is considered
degree zero. Products of letters are tricky. We add the degree of all the letters. So since
3a 2bc  3a 2b1c1 , the degree is 2+1+1 = 4. It is a 4 th degree term.
Try the following examples with your instructor. For each term, identify the numerical
coefficient and the variable part (if it has one). Also give the degree of the term.
Example 1: 8z 3
Example 2: r 2
Example 3: 15
Example 4: 11 b
One of the key things to know about terms is that we can only add or subtract terms with the
same variable part. So we can only add or subtract x with x and r 2 with r 2 and so on. Terms
with the same variable part are called “like terms.” To add or subtract like terms we add or
subtract the numerical coefficients and keep the variables (letters) the same.
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Look at the example of 5a+3a. Are these like terms? They both have the exact same letter
part, so they are like terms. Think of it like 5 apples plus 3 apples. We would have 8 apples, not
8 apples squared or 8 double apples. So 5a + 3a = 8a. We can combine the like terms and keep
the letter part the same.
Look at the example of 7a + 2b. Are these like terms? Since they have different letter parts,
they are not like terms. Hence we cannot add them. Think of it like 7 apples plus 2 bananas.
That will not equal 9 apple/bananas. It is just 7 apples and 2 bananas. That is a good way of
thinking about adding or subtracting terms that don’t have the same letter part. Hence 7a + 2b
= 7a + 2b. They stay separate. In fact, many formulas have two or more terms that cannot be
combined. 7a + 2b is as simplified as we can make it.
We even have special names for formulas that tell us how many terms it has. A formula with
only 1 term is called a “monomial”. A formula with exactly 2 terms is called a “binomial”.
A formula with exactly 3 terms is called a “trinomial”.
Try and simplify the following formulas with your instructor. After it is simplified, count how
many terms the simplified form has. Then name the formula as a monomial, binomial or
trinomial.
Example 5: 5w  8w
Example 7:

3 p   9q  5 p  4q
Example 6: 4m  9
Example 8: 6 x 2  8 x  14
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Practice Problems Section 5A
For each term, identify the numerical coefficient and the variable part (if it has one). Also give
the degree of the term.
2: 3r 4
1. 9L
3: 18
3
4. y
5: r 2
6: 12 p
7. 23v5
8: x 7
9: 52
10.  w2
11: 19h2
12: 3abc
2 2
13. 3x y
14: m2 n
15: vw2
16. 7b3
6
17: y
18: 17
8
19.  p
20: 19k 5
2
21: 13wxy
22. 19a3b2
4 2
23: p q
24: v 2 w2
Simplify the following formulas by adding or subtracting the like terms if possible. Count how
many terms the simplified form has and then name the formula as a monomial, binomial or
trinomial.
26. 12m  7m
27.
28. 5 x  14 x
29. 17m  9m  8m
30.
31. 3 y  8
32.
6a  4b  9b
33.
34. 8v  17v  12v
35.
4a  6b  8c
36. 3g  7h  10 g  5h
37. 5w  8 x  3 y
38.
x2  9 x2  7 x  1
39. 2w2  4w  8w
40. x3  7 x  9
41. 4m3  9m2  10m3  7m2
2
42. 5 y  7 y  8 y  3
25.

3a  11a

6v  14v

13 p   9 p  5 p

2 p   8 p  3m
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